Quantitative uniqueness for the power of the Laplacian with singular coefficients
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 3, pp. 513-529.

In this paper we study the local behavior of a solution to the $l$-th power of the Laplacian with singular coefficients in lower order terms. We obtain a bound on the vanishing order of the nontrivial solution. Our proofs use Carleman estimates with carefully chosen weights. We will derive appropriate three-sphere inequalities and apply them to obtain doubling inequalities and the maximal vanishing order.

Publié le :
Classification : 35J15,  35A02
@article{ASNSP_2011_5_10_3_513_0,
author = {Lin, Ching-Lung and Nagayasu, Sei and Wang, Jenn-Nan},
title = {Quantitative uniqueness for the power of the Laplacian with singular coefficients},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {513--529},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 10},
number = {3},
year = {2011},
zbl = {1237.35164},
mrnumber = {2905377},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2011_5_10_3_513_0/}
}
Lin, Ching-Lung; Nagayasu, Sei; Wang, Jenn-Nan. Quantitative uniqueness for the power of the Laplacian with singular coefficients. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 3, pp. 513-529. http://www.numdam.org/item/ASNSP_2011_5_10_3_513_0/

[1] G. Alessandrini, E. Beretta, E. Rosset and S. Vessella, Optimal stability for elliptic boundary value problems with unknow boundaries, Ann. Sc. Norm. Super. Pisa Cl. Sci. 29 (2000), 755–786. | Numdam | MR 1822407 | Zbl 1034.35148

[2] F. Colombini and C. Grammatico, Some remarks on strong unique continuation for the Laplacian and its powers, Comm. Partial Differential Equations 24 (1999), 1079–1094. | MR 1680873 | Zbl 0928.35041

[3] H. Donnelly and C. Fefferman, Nodal sets of eigenfunctions on Riemannian manifolds, Invent. Math. 93 (1988), 161–183. | EuDML 143593 | MR 943927 | Zbl 0659.58047

[4] N. Garofalo and F. H. Lin, Monotonicity properties of variational integrals, ${A}_{p}$ weights and unique continuation, Indiana Univ. Math. J. 35 (1986), 245–267. | MR 833393 | Zbl 0678.35015

[5] N. Garofalo and F. H. Lin, Unique continuation for elliptic operators: a geometric-variational approach, Comm. Pure Appl. Math. 40 (1987), 347–366. | MR 882069 | Zbl 0674.35007

[6] M. Giaquinta, “Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems”, Annals of Mathematics Studies, Vol. 105, Princeton University Press, NJ, 1983. | MR 717034 | Zbl 0516.49003

[7] R. Hardt and L. Simon, Nodal sets for solutions of elliptic equations, J. Differential Geom. 30 (1989), 505–522. | MR 1010169 | Zbl 0692.35005

[8] L. Hörmander “The Analysis of Linear Partial Differential Operators”, Vol. 3, Springer-Verlag, Berlin/New York, 1985. | MR 404822 | Zbl 0521.35002

[9] C. L. Lin, Strong unique continuation for $m$-th powers of a Laplaceian operator with singular coefficients, Proc. Amer. Math. Soc. 135 (2007), 569–578. | MR 2255304 | Zbl 1152.35117

[10] C. L. Lin, G. Nakamura and J.-N. Wang, Quantitative uniqueness for second-order elliptic operators with strongly singular coefficients, Rev. Mat. Iberoam. 27 (2011), 475–491. | MR 2848528 | Zbl 1219.35058

[11] F. H. Lin, Nodal sets of solutions of elliptic and parabolic equations, Comm. Pure Appl. Math. 44 (1991), 287–308. | MR 1090434 | Zbl 0734.58045

[12] M. Protter, Unique continuation for elliptic equations, Trans. Amer. Math. Soc. 95 (1960), 81–91. | MR 113030 | Zbl 0094.07901